Related papers: Bi-graded Koszul modules, K3 carpets, and Green's …
Green's canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the…
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence,…
We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length…
Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the…
We study the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic…
We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2.…
Green's conjecture is proved for smooth curves C lying on a rational surface S with an anticanonical pencil, under some mild hypotheses on the line bundle L defined by C. Constancy of Clifford dimension, Clifford index and gonality of…
We establish Green's syzygy conjecture for classes of covers of curves of higher Clifford dimension. These curves have an infinite number of minimal pencils, in particular they do not verify a well-known Brill-Noether theoretic sufficient…
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of…
Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's…
In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. We prove that if property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus $g$ then it…
We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field $k$. When $k=\CC$ this implies a conjecture of Green and Lazarsfeld.
We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.
Let S be a K3 surface and assume for simplicity that it does not contain any (-2)-curve. Using coherent systems, we express every non-simple Lazarsfeld-Mukai bundle on S as an extension of two sheaves of some special type, that we refer to…
A key result for syzygies of curves is Voisin's proof of Green's conjecture for the canonical embedding of a general curve of any genus. Her primary tools were the Lazarsfeld Mukai bundle on a K3 surface and a representation of Koszul…
Green's Conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin's results on syzygies of K3 sections, to the case of K3 surfaces with…
Inspired by the methods of Voisin, the first two authors recently proved that one could read off the gonality of a curve C from the syzygies of its ideal in any one embedding of sufficiently large degree. This was deduced from from a…
We generalize the modular Koszul duality of Achar-Riche to the setting of Soergel bimodules associated to any finite Coxeter system. The key new tools are a functorial monodromy action and wall-crossing functors in the mixed modular derived…
We extend the notion of Clifford index to reduced curves with planar singularities by considering rank 1 torsion free sheaves. We investigate the behaviour of the Clifford index with respect to the combinatorial properties of the curve and…
It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in $M_0\otimes V$…